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11: 4.37 Inverse Hyperbolic Functions

… ► $\mathrm{Arcsinh}z$ and $\mathrm{Arccsch}z$ have branch points at $z=\pm \mathrm{i}$; the other four functions have branch points at $z=\pm 1$. … ►

§4.37(iv) Logarithmic Forms

… ►

Other Inverse Functions

►For the corresponding results for $\mathrm{arccsch}z$, $\mathrm{arcsech}z$, and $\mathrm{arccoth}z$, use (4.37.7)–(4.37.9); compare §4.23(iv). …

12: 22.4 Periods, Poles, and Zeros

… ►Again, one member of each congruent set of zeros appears in the second row; all others are generated by translations of the form $2mK+2n\mathrm{i}{K}^{\prime }$, where $m,n\in \mathbb{Z}$. …

13: 34.2 Definition: $\mathit{3}j$ Symbol

… ►Either all of them are nonnegative integers, or one is a nonnegative integer and the other two are half-odd positive integers. They must form the sides of a triangle (possibly degenerate). … ►For alternative expressions for the $\mathit{3}j$ symbol, written either as a finite sum or as other terminating generalized hypergeometric series ${}_3{}^{}F_2^{}$ of unit argument, see Varshalovich et al. (1988, §§8.21, 8.24–8.26).

14: 31.12 Confluent Forms of Heun’s Equation

… ►This has regular singularities at $z=0$ and $1$, and an irregular singularity of rank 1 at $z=\mathrm{\infty }$. …

15: 14.5 Special Values

… ►

§14.5(v) $\mu =0$, $\nu =\pm \frac{1}{2}$

… ► ►

14.5.20 $${𝖯}_{\frac{1}{2}}\left(\cos \theta \right)=\frac{2}{\pi }\left(2E\left(\sin \left(\frac{1}{2}\theta \right)\right)-K\left(\sin \left(\frac{1}{2}\theta \right)\right)\right),$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $E\left(k\right)$: Legendre’s complete elliptic integral of the second kind, $\cos z$: cosine function, ${𝖯}_{\nu }\left(x\right)={𝖯}_{\nu }^0\left(x\right)$: Ferrers function of the first kind, $\sin z$: sine function and : variable

A&S Ref:

8.13.11 (in slightly different form)

Permalink:

http://dlmf.nist.gov/14.5.E20

Encodings:

pMML, png, TeX

See also:

Annotations for §14.5(v), §14.5 and Ch.14

… ►

14.5.22 $${𝖰}_{\frac{1}{2}}\left(\cos \theta \right)=K\left(\cos \left(\frac{1}{2}\theta \right)\right)-2E\left(\cos \left(\frac{1}{2}\theta \right)\right),$$

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Symbols:

$K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $E\left(k\right)$: Legendre’s complete elliptic integral of the second kind, $\cos z$: cosine function, ${𝖰}_{\nu }\left(x\right)={𝖰}_{\nu }^0\left(x\right)$: Ferrers function of the second kind and : variable

A&S Ref:

8.13.12 (in slightly different form)

Permalink:

http://dlmf.nist.gov/14.5.E22

Encodings:

pMML, png, TeX

See also:

Annotations for §14.5(v), §14.5 and Ch.14

►

14.5.23 $${𝖰}_{-\frac{1}{2}}\left(\cos \theta \right)=K\left(\cos \left(\frac{1}{2}\theta \right)\right).$$

ⓘ

Symbols:

$K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $\cos z$: cosine function, ${𝖰}_{\nu }\left(x\right)={𝖰}_{\nu }^0\left(x\right)$: Ferrers function of the second kind and : variable

A&S Ref:

8.13.10 (in slightly different form)

Permalink:

http://dlmf.nist.gov/14.5.E23

Encodings:

pMML, png, TeX

See also:

Annotations for §14.5(v), §14.5 and Ch.14

…

16: 18.40 Methods of Computation

… ►Orthogonal polynomials can be computed from their explicit polynomial form by Horner’s scheme (§1.11(i)). Usually, however, other methods are more efficient, especially the numerical solution of difference equations (§3.6) and the application of uniform asymptotic expansions (when available) for OP’s of large degree. For applications in which the OP’s appear only as terms in series expansions (compare §18.18(i)) the need to compute them can be avoided altogether by use instead of Clenshaw’s algorithm (§3.11(ii)) and its straightforward generalization to OP’s other than Chebyshev. …

17: 18.7 Interrelations and Limit Relations

§18.7 Interrelations and Limit Relations

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Chebyshev, Ultraspherical, and Jacobi

… ►

Legendre, Ultraspherical, and Jacobi

… ►

§18.7(iii) Limit Relations

… ► See §18.11(ii) for limit formulas of Mehler–Heine type.

18: 4.23 Inverse Trigonometric Functions

… ► $\mathrm{Arctan}z$ and $\mathrm{Arccot}z$ have branch points at $z=\pm \mathrm{i}$; the other four functions have branch points at $z=\pm 1$. … ►

§4.23(iv) Logarithmic Forms

… ►

Other Inverse Functions

… ►Care needs to be taken on the cuts, for example, if then $1/(x+\mathrm{i}0)=(1/x)-\mathrm{i}0$. …

19: 1.9 Calculus of a Complex Variable

… ►or in polar form (1.9.3) $u$ and $v$ satisfy … ►One of these domains is bounded and is called the interior domain of $C$; the other is unbounded and is called the exterior domain of $C$. … ►or its limiting form, and is invariant under bilinear transformations. ►Other names for the bilinear transformation are fractional linear transformation, homographic transformation, and Möbius transformation. …

20: 31.11 Expansions in Series of Hypergeometric Functions

… ►For other expansions see §31.16(ii). ►

§31.11(ii) General Form

… ►The expansion (31.11.1) for a Heun function that is associated with any branch of (31.11.2)—other than a multiple of the right-hand side of (31.11.12)—is convergent inside the ellipse $ℰ$. …